I was just in Turkey in September-October (I eventually wrote up my whole trip on TravelPod if you want to have a look) and while in Istanbul we stayed at a place called Atlantis Hotel in Sultanahmet. It was 55 Euros for a double bed, 35 for a single. I rather enjoyed it there; the rooftop terrace didn’t have a very spectacular view I suppose but the location was fantastic. We were an easy walk to the Blue Mosque, Hagia Sophia, the Basilica Cistern, Topkapi Palace, Grand Bazaar, and Spice Bazaar, and the Sultanahmet tram stop was at the base of the street the hotel was on.
OTTOMH, I spent about 4 hours at Topkapi palace and didn’t see enough (just the Treasury and the Harem and the palace grounds). Could do a whole day there easily, especially if you wanted to take a guided tour. 2 hours at Hagia Sophia (half of it is currently blocked up with scaffolding), 2 hours at Blue Mosque, a whole afternoon at the Grand Bazaar, an hour or so at the Spice Bazaar.
For other things to do in Istanbul, might I suggest the following:
Check out Biletix.com and try to see a concert/play in Aya Irini - I think it’s the oldest standing building in Istanbul, it’s a little church in the Topkapi palace grounds. The acoustics are fantastic and the somewhat barren and ascetic-looking church really gives you a break from having colourful mosaics and frescoes leaping out at you from every available surface like so many other churches in the city. [2 hrs]
The Istanbul Archaeological Museum (also on Topkapi palace grounds) is a must-see, especially the Museum of the Ancient Orient. It has an amazing collection. [3-4 hrs depending on how thorough you are in a museum. Has a childrens’ museum in it too if that is of any use to you.]
A walk up Istiklal Caddesi in Beyoglu, in the evening, culminating in Taksim Square as previously recommended, is a great way to spend a night. Lots of little book stores, quirky shops, restaurants, and a good night scene. There’s also a church here that holds concerts as well, but I can’t tell you much about it 'cause I passed on it.
A cruise on the Bosphorus is a great idea, but I would recommend against taking an organised tour (especially if it’s SheTours). I ended up paying 65 Euros for something that really should have only cost 1/4 to 1/3 of that, including all of our travel on public transportation. I’d take the public ferries up and down the Bosphorus, or possibly head over to the Asian side and take ferries out to the Princes’ Islands in the Sea of Marmara if you want to make a day of it.
I had a great time at Miniaturk (take the Bosphorus ferry to Sutluce and it’s a 20-min walk from there) - but that was at the end of 5 weeks of hardcore museum- and ruins-faring travel. It’s a silly little place with miniature statues of all the sights to see around Turkey. It’s kinda cool 'cause it shows what the sites would have looked like in their prime, rather than ravaged by age and the British Museum
[1 1/2 - 2 hrs]
Chora Church in Edirnekapi is beautiful, and right next door there is a restaurant that serves authentic medieval Ottoman recipes from Topkapi palace. Gimmicky and expensive? Sure, but also a really cool experience. [1 hr]
Grand Palace Mosaic Museum in the bazaar behind the Blue Mosque is pretty cool too. Best mosaics I saw in my whole trip
[1 hr]
Really, you can’t go wrong in Istanbul, it’s full of awesome.
As for day trips, Edirne is nice, Troy/Gallipoli is pretty good but Troy is mostly not-there, so if you can swing a trip out to Ephesos for some seriously awe-inspiring ruins you should give that a try. I might also recommend Pamukkale if you can get a tour that will take you to Aphrodisias 30km away - Aphrodisias is as amazing as Ephesos but fewer people are there. (When I was in Ephesos there must have been, I don’t know, 5,000 other people there?)
Hope some of this information helps!